Amorphous carbon and its characteristics

Generally we can characterize the amorphous structures by the high degree of short range order and absence of long range order. From the energetic point of view, atoms in an amorphous crystal are not bonded ideally, they are subject to important stresses and distortions. The energy of an amorphous solid is thus higher than that of a pure crystal.

There are two specific amorphous form of carbon: the diamond-like amorphous carbon ($ta-C$) and the graphite-like amorphous carbon ($a-C$). These two structures can be distinguished clearly by their macroscopic and microscopic properties. The former has higher density, is transparent and much harder than the latter. From the microscopic point of view, the ratio of fourfold, diamondlike bonds to threefold, graphite-like bonds ($sp^3/sp^2$) will determine the kind of structure we obtain. This ratio is strongly affected by the way the amorphous solid is prepared and depends on temperature and pressure.

In order to describe an amorphous structure the following characteristics can be used: a coordination number, a radial distribution function, an angular distribution function. The coordination number $z$ is the number of nearest neighbor atoms. For example, $z$ is 4 for the diamond structure, or 12 for the FCC structure. For perfect lattices, the coordination number has no real significance but for more complex structures, like amorphous lattices, it plays a crucial role in the determination of the amorphous structure type.

The radial distribution function $g(r)$ is a generalization of the coordination number. Instead of looking at the first nearest neighbors only, one now counts the number of atoms that lie at the distance $r$ from a specific atom, averaging over all the atoms of the lattice. When normalized $g(r)$ is precisely the probability of finding a neighboring atom at distance $r$. It is clear that for a perfect lattice, $g(r)$ will give delta functions at characteristic distances of the lattice. The $g(r)$ function, as a coordination number, can be very useful for a description of more complicated structures. For example, short-range order is expressed by one of two distinct and broad peaks in the shortest distances, following by a quite flat tail, which is characteristic to the $g(r)$ of amorphous structure. For the $a-C$ structure, for instance, the first peak is centered near the graphite bond length (1.42 Å) and is broad enough to include the diamond bond length (1.54 Å), so that many bonds, in the graphite-like structure, can be specified as diamond-like bonds (see Fig.2.4). The liquid phase exhibits a very similar form, except that the peaks are broader and shallower than in the amorphous case [8].

The bond angle distribution function $g(\theta )$ is defined for angles between nearest neighbors atoms. For a diamond crystal, $g(\theta )$ is a delta function centered at $\theta=109.47^{\circ}$. For an amorphous crystal, $g(\theta )$ is centered at an angle close to the tetrahedral angle for the $ta-C$ structure and to $\theta=120^{\circ}$ for the $a-C$ structure. Large angle distortions occurs in these structures, as is indicated by the significant width of the bond angle distribution.

Figure 2.4: $g(r)$ for an $a-C$ sample (up), taken from ref. [6] and $g(\theta )$ for a $ta-C$ sample (down), taken from ref. [7]. A$_4$ and A$_3$ are the contribution of the fourfold and the threefold atoms respectively.